Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x-7y &= 1 \\ 5x+9y &= 3\end{align*}$
Answer: Begin by moving the $x$ -term in the second equation to the right side of the equation. $9y = -5x+3$ Divide both sides by $9$ to isolate $y$ $y = {-\dfrac{5}{9}x + \dfrac{1}{3}}$ Substitute this expression for $y$ in the first equation. $-5x-7({-\dfrac{5}{9}x + \dfrac{1}{3}}) = 1$ $-5x + \dfrac{35}{9}x - \dfrac{7}{3} = 1$ Simplify by combining terms, then solve for $x$ $-\dfrac{10}{9}x - \dfrac{7}{3} = 1$ $-\dfrac{10}{9}x = \dfrac{10}{3}$ $x = -3$ Substitute $-3$ for $x$ back into the top equation. $-5( -3)-7y = 1$ $15-7y = 1$ $-7y = -14$ $y = 2$ The solution is $\enspace x = -3, \enspace y = 2$.